We outline a general formalism of hydrodynamics for quantum systems with multiple particle species which undergo completely elastic scattering. In the thermodynamic limit, the complete kinematic data of the problem consist of the particle content, the dispersion relations, and a universal dressing transformation which accounts for interparticle interactions. We consider quantum integrable models and we focus on the one-dimensional fermionic Hubbard model. By linearizing hydrodynamic equations, we provide exact closed-form expressions for Drude weights, generalized static charge susceptibilities, and charge-current correlators valid on the hydrodynamic scale, represented as integral kernels operating diagonally in the space of mode numbers of thermodynamic excitations. We find that, on hydrodynamic scales, Drude weights manifestly display Onsager reciprocal relations even for generic (i.e., noncanonical) equilibrium states, and establish a generalized detailed balance condition for a general quantum integrable model. We present exact analytic expressions for the general Drude weights in the Hubbard model, and explain how to reconcile different approaches for computing Drude weights from the previous literature.
We outline a general formalism of hydrodynamics for quantum systems with multiple particle species which undergo completely elastic scattering. In the thermodynamic limit, the complete kinematic data of the problem consist of the particle content, the dispersion relations, and a universal dressing transformation which accounts for interparticle interactions. We consider quantum integrable models and we focus on the one-dimensional fermionic Hubbard model. By linearizing hydrodynamic equations, we provide exact closed-form expressions for Drude weights, generalized static charge susceptibilities, and charge-current correlators valid on the hydrodynamic scale, represented as integral kernels operating diagonally in the space of mode numbers of thermodynamic excitations. We find that, on hydrodynamic scales, Drude weights manifestly display Onsager reciprocal relations even for generic (i.e., noncanonical) equilibrium states, and establish a generalized detailed balance condition for a general quantum integrable model. We present exact analytic expressions for the general Drude weights in the Hubbard model, and explain how to reconcile different approaches for computing Drude weights from the previous literature.
Building on the Bethe ansatz and generalized hydrodynamics, we rigorously establish universal relations between Drude weights governing charge, spin, and energy transport and thermodynamic properties for the one-dimensional repulsive two-component continuum Fermi gas - the paradigmatic integrable Yang-Gaudin model. We analytically derive thermodynamic expressions for charge Drude weights: Dnn, Dnm, Dne (responses of particle, magnetization and energy currents to a chemical potential gradient) equal the particle density, magnetization, and temperature times entropy density, respectively. These relations constitute a hallmark of ballistic transport, and can be generalized to other 1D continuum integrable systems. Furthermore, we investigate spin Drude weight (Dmm, Dme) at zero and low temperatures. They characterize the magnetization and energy current responses to a magnetic field gradient, revealing an essential spin-charge coupling feature in quantum transport. We find that the Drude weights Dnm, Dmm, and Dme across the μ-H plane remarkably map out the zero-temperature phase boundary. This work fills a critical gap in the understanding of transport within integrable quantum gases and furnishes a direct theoretical foundation for future ultracold-atom experiments.
The Drude weight characterizes ballistic transport in quantum many-body systems. Although analytical calculations of Drude weights have been extensively studied in integrable models, their direct connections to finite-temperature macroscopic state functions remain unestablished, especially for continuous multicomponent quantum gases. In the present work, we use generalized hydrodynamics and the thermodynamic Bethe ansätz to calculate exactly the Drude weights for one-dimensional continuous integrable systems, including the Lieb-Liniger and Bose-Fermi mixture models. We derive universal exact relations between Drude weight matrix components and key thermodynamic densities (particle density, enthalpy, entropy). Analytic expressions for Drude weight are obtained across different physical regimes, i.e. strong- and weak-coupling regimes in addition to universal scaling laws near the quantum phase transitions. To bridge theory and experiment, we simulate two experimental protocols, linear potential quench and bipartitioning quench, to enable reliable measurements of the Drude weights. Using these protocols, we calculate the charge and energy Drude weight for Lieb-Liniger gas and compare with recent measurements reported in [Science 391, 290 (2026)], showing excellent agreement with particle density and enthalpy, respectively, thus offering deeper physical insights into experimental observations. Our findings directly link ballistic transport properties to thermodynamics, providing rigorous theoretical benchmarks for future ultracold atomic gas experiments.
Understanding charge transport in strongly correlated systems remains a central challenge in condensed matter physics, while exact analytical results for finite-temperature transport remain rare. Here, we investigate charge transport in the one-dimensional Hubbard model in the infinite-interaction limit. We first construct the exact and explicit, i.e., beyond Bethe ansatz energy spectrum and eigenstates. Building on these explicit eigenstates, we derive a closed-form analytical expression for the charge Drude weight at arbitrary temperatures in the hole-dilute limit with a fixed number of doped holes, as well as analytical expressions in the low- and high-temperature limits at fixed hole density. We further show that, at low temperatures, the leading correction to the charge Drude weight is linear in T when the number of holes is fixed, whereas it is quadratic in T when the hole density is fixed. This analytically controlled example highlights the importance of identifying the underlying thermodynamic trajectory when interpreting finite-size numerical studies of charge transport in more general models.
Hydrodynamic projections, the projection onto conserved charges representing ballistic propagation of fluid waves, give exact transport results in many-body systems, such as the exact Drude weights. Focussing one one-dimensional systems, I show that this principle can be extended beyond the Euler scale, in particular to the diffusive and superdiffusive scales. By hydrodynamic reduction, Hilbert spaces of observables are constructed that generalise the standard space of conserved densities and describe the finer scales of hydrodynamics. The Green-Kubo formula for the Onsager matrix has a natural expression within the diffusive space. This space is associated with quadratically extensive charges, and projections onto any such charge give generic lower bounds for diffusion. In particular, bilinear expressions in linearly extensive charges lead to explicit diffusion lower bounds calculable from the thermodynamics, and applicable for instance to generic momentum-conserving one-dimensional systems. Bilinear charges are interpreted as covariant derivatives on the manifold of maximal entropy states, and represent the contribution to diffusion from scattering of ballistic waves. An analysis of fractionally extensive charges, combined with clustering properties from the superdiffusion phenomenology, gives lower bounds for superdiffusion exponents. These bounds reproduce the predictions of nonlinear fluctuating hydrodynamics, including the Kardar-Parisi-Zhang exponent 2/3 for sound-like modes, the Levy-distribution exponent 3/5 for heat-like modes, and the full Fibonacci sequence.
Owing to the fact that the particle current operator in nonrelativistic gases is proportional to the total momentum operator, the particle transport in such systems is always ballistic and fully characterized by a Drude weight Δ. The Drude weight can be calculated within linear response theory. It is given by the formula Δ=2πD, where D is the density of the gas. This holds in any dimension and for every equilibrium ensemble, in particular for generalized Gibbs ensembles that describe possible equilibrium states of isolated integrable quantum systems. In the canonical ensemble case, the Drude weight can be equivalently obtained from a generalized susceptibility related to the fluctuations of the conserved particle current. Such susceptibility can be rigorously calculated for the integrable Lieb-Liniger Bose gas in any generalized Gibbs ensemble using a generalized Yang-Yang thermodynamic formalism. The resulting expression agrees with a prediction made within the context of generalized hydrodynamics. It also allows us to see explicitly that, within truly generalized Gibbs ensembles, the conductivity related with the particle current is not determined by the corresponding current-current autocorrelation function.
Abstract We review the recent advances on exact results for dynamical correlation functions at large scales and related transport coefficients in interacting integrable models. We discuss Drude weights, conductivity and diffusion constants, as well as linear and nonlinear response on top of equilibrium and non-equilibrium states. We consider the problems from the complementary perspectives of the general hydrodynamic theory of many-body systems, including hydrodynamic projections, and form-factor expansions in integrable models, and show how they provide a comprehensive and consistent set of exact methods to extract large scale behaviours. Finally, we overview various applications in integrable spin chains and field theories.